EPJ Web of Conferences 21, 08002 (2012)
DOI: 10.1051/epjconf/20122108002
C Owned by the authors, published by EDP Sciences, 2012
Fission via compound states and J π K A. Bohr’s channels:
what we can learn from recent studies with slow neutrons
A.L. Barabanov1,a and W.I. Furman2
1
2
NRC ”Kurchatov Institute”, Moscow 123182, Russia; Moscow Institute of Physics and Technology,
Dolgoprudny 141700, Moscow Region, Russia
Frank Laboratory of Neutron Physics, JINR, Dubna 141980, Russia
Abstract. Last data on angular correlations of fission fragments from slow (s-wave)
neutron induced binary fission of spin-aligned nuclei 235 U are discussed in the context of
J π K A. Bohrs channels. Special attention is paid to K = 0 channel. Reasons for its suppression are specified for compound nucleus states of negative parity. A brief overview of
recent data on T-odd angular correlations in ternary and binary (with emission of a third
particle, a neutron or γ-quantum) fission induced by slow polarized neutrons is presented.
On the basis of the developed theoretical approach it is shown that a valuable information on J π K fission channels at scission point can be inferred from these T-odd angular
correlations.
1 Introduction
Fission of nuclei by slow neutrons is of great interest due to a possibility of studying of the process’s quantum aspects related to fission channels. Really, modern high resolution time-of-flight technique makes possible an identification of quantum interference of amplitudes of fission via separated
compound-states (neutron resonances) both in differential and in total cross sections of (n,f)-reaction.
z
J
p1
6
*
6
g
A
A
K
k
p2
Fig. 1. Fission of a nucleus with spin J into two fragments; K is the projection of spin J on the fission axis.
More exactly, the fission amplitudes are related to channels introduced by A. Bohr [1]. Each channel is characterised not only by spin J and parity π of the corresponding compound-state, but also by
a projection K of spin J onto an axis n f of nuclear elongation (see Fig. 1). In fact, the channels are
nuclear rotation states (transition states), and K is a good quantum number due to conservation of axial
symmetry of the fissioning nucleus up to scission.
The A. Bohr’s model allows to connect an angular distribution of fission fragments with a distribution over the states with different K on a fission barrier. Indeed, the wave function of the deformed
nucleus is
X
X
ΨJ ∼
a M (J)
g JK ΦK (τ)D JMK (n f ),
(1)
M
a
K
e-mail: [email protected]
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EPJ Web of Conferences
where M is a projection of spin J on an axis z, and the function ΦK (τ) describes internal degrees of
freedom of deformed nucleus. Thus, the distribution over n f takes the form
Z
X
dw(n f )
3 cos2 θ − 1
∼
|Ψ J |2 dτ ∼ 1 + 3 p2 hA2 i
+ . . . , hA2 i =
|g JK |2 A2 (K),
(2)
dΩ
2
K
where p2 is the parameter of spin alignment of nuclei with respect to the axis z. The factor
A2 (K) ∼
3K 2
−1
J(J + 1)
(3)
depends strongly on K, therefore the coefficient of angular anisotropy hA2 i is very sensitive to the
distribution |g JK |2 over K. However, the A. Bohr’s model incorporates no fragment characteristics.
Thus, any predictions for fragment characteristics, for example, for the fragment spin alignment, are
impossible.
z
6
L
J1
KA
A
A p1
*
g
K
1
k
@AK2
R
J2@
p2
Fig. 2. Two fragments with spins J1 and J2 , helicities K1 and K2 and momenta p1 and p2 ; L is a relative orbital
momentum of two fragments.
Another approach for the angular distribution of fission fragments was proposed by Strutinsky [2]
in the frame of a so-called helicity representation (see Fig. 2). After the neck rupture, the spin J
transfers into J1 +J2 +L, where J1 and J2 are the fragment spins and L is the relative orbital momentum
of two fragments. Therefore, the projection K transfers near-precisely into a total helicity of two fission
fragments K̃ that is the projection of total spin F = J1 + J2 of the fragments onto the fission axis n f
(notice that L ⊥ n f ). According Strutinsky, K̃ is also the good quantum number in the fission process
due to insignificance of the relative orbital momentum of the fragments. As A. Bohr, Strutinsky has
considered only a simplified case of fission of a nucleus with definite spin and parity J π .
The Strutinsky’s method deals not only with the fragment’s helicities but with all characteristics
of fragments in the exit states labeled by an index α. The asymptotic wave function in the (LF)representation is
ΨJ →
X
eikα r X
a M (J) (−i)L+1 gα (LF) ϕαLF JM ,
r M
LF
ϕαLF JM =
X
JM
C FνLm
χαFν iL YLm (nα ),
(4)
νm
where ϕαLF JM is the channel function. The fragment’s functions can be expressed via the functions with
definite helicities,
X
F
χαFν =
DνK
(nα )χαFK .
(5)
K
Thus, the total wave function in the (FK)-representation takes the form (see details in [8, 9])
ΨJ →
X
eikα r X
a M (J)
gα (FK) ϕαFK JM ,
r M
FK
ϕαFK JM = χαFK DFMK (nα ),
(6)
where ϕαFK JM is the channel function in the helicity representation. Notice that the right-hand side
of Eq. (6) looks like the right-hand side of Eq. (1). Thus, the Strutinsky’s method leads to the same
form (2) for the angular distribution of fission fragments. In addition, this approach allows to describe,
in principle, all observable quantities after scission point. In particular, spin polarization and alignment
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parameters for fission fragments were expressed for the first time in [3, 4] via amplitudes of fission to
states with given F and K and then numerically estimated.
Later the Strutinsky’s method was extended in [5–8] to consistent description of differential crosssection of (n,f)-reaction with polarized neutrons and spin-aligned target nuclei (see also [9]). We have
developed a method of summation over numerous exit states and related these states to A. Bohr fission
channels. The fission amplitudes
γνα (FK J π ) ∼ hϕαFK JM |XνJπM i,
(7)
where XνJπM is a wave function of the compound resonance ν, can be simplified with the use of the
assumption by Furman and Kliman [10]
X
X
JπM
XνJπM = Xνc
+ XνJπM
XνJπM
=
αmK ΨmJπK M .
(8)
aνJπK
f ,
f
f
K
m
JπM
Here the functions Xνc
and XνJπM
correspond to small and large deformations. The latter is presented
f
as a superposition on the functions ΨmJπK M ∼ ΦmK (τ)D JMK (n f ) (see Eq. (1)). Each of these functions
depends on fission mode m and the projection K but does not depend on resonance’s index ν. Only the
amplitudes aνJπK
f describing the mixing over K by Coriolis interactions keep the dependence on ν. This
fact leads to survival of interference terms between different compound resonances at summation over
all exit states α. As a result, the fission amplitudes γν (J π K) for A. Bohr’s channels arise. By this way
direct connection was established between the J π K A.Bohr’s channels and all observable quantities in
the fission process.
The most reliable source of information on A. Bohr’s J π K channels are the angular distributions
of fragments from fission of compound states with definite J π , i.e. neutron resonances. In order that
compound nuclei arising after capture of s-wave neutrons are spin aligned, an ensemble of target nuclei
should possess the spin alignment. The most thorough studies were performed for the system n+235 U.
In the late 1960s and in the early 1970s the coefficient of angular anisotropy has been measured in
separated resonances [11,12], and then in 1990s in Dubna the angular anisotropy has been studied as
function of neutron energy in resonances and between them [13, 14]. Analysis of these data provides
the most complete information on fission channels with various K. Recently we have shown [15] that
the channel K = 0 is especially interesting because its contribution is very sensitive to the shape
symmetry of the fissioning nucleus at first and second fission barriers. Below, in the second section of
the report, it is explained in more details.
During the last decade new angular correlations, known as T (formal, not real, violation of Time
reversal invariance) and R (Rotation) effects [16–20], have been found in fission of nuclei 233 U and
235
U by slow polarized neutrons. Formally both angular correlations are odd with respect to time
reverse (T-odd), but they have nothing to do with violation of fundamental time reversal invariance.
Both correlations are caused by specific interactions in the final states. The third section of the report is
devoted to discussion of these T-odd correlations. Some of them give us an estimation for the accuracy
of the relation K̃ = K crucial for the theoretical approache based on helicity representation. We argue
that the observed smallness of R effect points out to high accuracy of the above equality.
2 Structural features of the fission channel K = 0
In processing data from experiments [13, 14], special attention was paid to the K = 0 channel. At
the first stage of data processing [13] only the J π K = 3− 0 channel was taken into account while the
J π K = 4− 0 channel was excluded according the original simplified model [1]. Later the J π K = 4− 0
channel was included to the analysis (see below for the reasons). The quantitative estimate of the
contribution of the K = 0 channels has been obtained for the first time. This contribution is proved
to be sizable (on the scale of 25% !). However, the K = 0 channel does not dominate over K , 0
channels. The suppression of the K = 0 channel was noted already in [12] where it was interpreted as
a disagreement with A. Bohr’s model [1].
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In fact, we should be guided not by the original simplified model [1] but by the extended approach
to deformed nuclei presented in the monography [21]. According this approach, both channels, J π K =
3− 0 and 4− 0, are open (see details in [15]). But the wave function of the deformed nucleus with K = 0
is characterized by an additional quantum number, s = ±1, that is an eigenvalue of the operator Ŝ i
of inversion of the internal wave function with respect to a plane that contains the axial-symmetry
axis. At the R-symmetric stage, this wave function is characterized by an additional quantum number,
r = ±1, that is an eigenvalue of the operator R̂i of rotation on π of the internal wave function around the
axis that is orthogonal to the axial-symmetry axis and which passes through the center of the nucleus
being considered. The quantum numbers r, s, and π are related by the equality
r = sπ.
(9)
The nucleus 235 U is of negative parity, thus π = −1 for resonances excited by s-wave neutrons.
One can readily see that the equality r = −s always holds; that is, either r or s takes the value of −1.
For the 235 U nucleus the first barrier appears R-symmetric, while the second one is R-asymmetric. This
means that on any barrier, for any value of the spin J, the internal function changes its sign under one
transformation or another, thus the corresponding internal quantum state possesses an enlarged energy.
Therefore, the K = 0 channel is suppressed. In particular, it is the case in the fission of spin-aligned
235
U nuclei of negative-parity by slow (s-wave) neutrons.
3 T-odd angular correlations in fission with a third emitted particle
x
6
1
2
α-particle BM
s
y
pα
BM s z
iθ B m
pLF
light
heavy
fragment fragment
r
neutron
3
4
Fig. 3. Scheme of registration of T and R effects. The neutron spin s is directed along the axis y, the momentum of
the light fragment pLF is directed along the axis z, the asymmetry of a third particle’s emission along and against
to the axis x is under investigation.
Scheme of measurement of small (∼ 10−3 ) asymmetries of third particle’s emission in nuclear
fission induced by slow (s-wave) polarized neutrons is presented on Fig. 3. The incident neutrons are
longitudinally polarized. The neutrons move along the axis y, the light fragments move along the axis
z, then there are asymmetries of third particle’s detection, e.g. α-particle in ternary fission, by pairs of
counters 1 and 3 or 2 and 4. Let us introduce asymmetry coefficients
D13 (θ) =
N1 − N3
,
N1 + N3
D24 (π − θ) =
N2 − N4
,
N2 + N4
(10)
and total asymmetry
D=
(N1 + N2 ) − (N3 + N4 ) D13 + λD24
=
,
(N1 + N2 ) + (N3 + N4 )
1+λ
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λ=
N2 + N4
,
N1 + N3
(11)
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where λ is of the scale of unity (for ternary fission with α-particle’s emission, λ is smaller than unity,
because the maximum of α-particle’s angular distribution falls to the angle θ ' 830 due to a greater
Coulomb repulsion from the heavy fragment). Obviously, D13 = D24 = 0 for unpolarized neutrons.
In the experiments [16,17] with target nuclei 233 U it was found that D13 ' D24 ' D. Formally,
it corresponds to 3-fold correlation (pα [s × pLF ]), which is T-odd. Thus the asymmetry was named
the T effect. In [22] a qualitative explanation of the phenomenon as the result of the interaction of the
spin of residual nucleus and the orbital momentum of α-particle (spin-orbit interaction) was proposed.
However, this hypothesis was met with a scepticism based on naive semiclassical interpretation of
spin-orbit interaction. It was noted in [17]: ”. . . in case of the spin-orbit interaction is at work, the
correlation coefficient D should have opposite signs for angles smaller or larger that the average angle
830 . . . ”, i.e. the relation D13 ' −D24 (and D ' 0) was expected.
Surprisingly, just this result D13 ' −D24 (and D ' 0) has been found in the middle of 2000s in
ternary fission of target nuclei 235 U by slow polarized neutrons [18]. However, another semiclassical
model for the source of this asymmetry was proposed based on the rotation of the fissioning nucleus.
Thus, the corresponding asymmetry was named the R (Rotation) effect. Formally, it corresponds to
5-fold correlation (pα [s × pLF ])(pα pLF ) (which is also T-odd).
α-particle s
BM
ραB
i
light
fragment
B
ρf
m
heavy
fragment
Fig. 4. Two fragments and α-particle arise after ternary fission.
The approach [22] can be extended to the use of hyperspherical harmonics to describe 3 particles
in the final state of ternary fission (see Fig. 4). It gives for the double differential fission cross section:
X
dσ
τQ0 (s) A(Q, Λ f , Λα ) φΛQf Λα (n s , nLF , nα ),
=
dΩ f dΩα Q,Λ ,Λ
f
(12)
α
where φΛQf ,Λα (n s , nLF , nα ) are invariant spherical functions of unit vectors n s = s/s, nLF = ρ f /ρ f ,
nα = ρα /ρα (see [23,9])
X
∗
φΛQf Λα (n s , nLF , nα ) = (4π)3/2
YQq
(n s ) YΛ f λ f (nLF ) YΛα λα (nα ).
(13)
q,λ f ,λα
Here τQ0 (s) (Q = 0,√
1) is the spin-tensor of orientation for the incident neutrons, therewith τ00 (s) = 1
and τ10 (s) = p(s)/ 3, where p(s) is the neutron polarization; A(Q, Λ f , Λα ) is a cumbersome factor containing a bilinear combination of the S-matrix elements corresponding to transitions from
the entrance neutron channel to the exit fission channels. It is easy to see that the terms in (12) for
Q = Λ f = Λα = 1 and Q = 1, Λ f = Λα = 2 describe the observed angular correlations in ternary
fission. Indeed,
φ111 (n s , nLF , nα ) ∼ (nα [n s × nLF ]),
φ122 (n s , nLF , nα ) ∼ (nα [n s × nLF ])(nα nLF ).
(14)
Thus, in the proposed approach the 3-fold and 5-fold correlations (T and R effects) in the neutron
induced ternary fission are caused by interference of exit channels with different schemes of angular
momenta’s summation. Survival of the interference effects at summations over all exit states may be
due to simplicity of the spin-orbit interaction in the system of two fragments and α-particle.
The other mechanism is responsible for formation of T-odd angular correlations in binary fission
where the third particle is a γ-quantum or a neutron emitted by one of fragments (see Fig. 5). Such
correlations was discovered in [19,20]. In a sequential process (first form the fragments, and then one
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γ-quantum s
pγBMB
J
pf m
Bi
light
fragment
heavy
fragment
Fig. 5. Two fragment arise after binary fission of polarized nucleus with spin J, while a third particle, e.g. a
γ-quantum, is emitted by one of the fragments.
of the fragments emits the third particle) the 3-fold T-odd asymmetry (nT P [n s × nLF ]) (T effect) is
suppressed by a double parity prohibition (i.e. is practically zero) [24]; here nT P is the unit vector
along the direction of movement of the third particle. But the 5-fold T-odd asymmetry (R effect),
(nT P [n s × nLF ])(nT P nLF ), can exist.
The 5-fold correlation is bilinear in nT P , therefore the fragment’s spin alignment is needed to produce the correlation. In the fission process the fragments are really spin aligned. Thus, the mechanism
should be only established how the polarization of the fissioning nucleus with spin J influences the
fragment’s spin alignment. This problem as noted above has been considered for the first time in [4]
(see also [9]).
The differential probability of the third particle’s emission, e.g. the γ-quantum, takes the form
X
dw
H
=
τQ0 (s) B(Q, Λ, H) φΛQ
(nγ , nLF , n J ),
dΩγ Q,Λ,H
(15)
where nγ = pγ /pγ , nLF = pLF /pLF , n J = J/J. Here τQ0 (J) is the spin-tensor of orientation of the
√
fissioning nucleus, therewith τ00 (J) = 1 and τ10 (J) = p(J) J/(J + 1), where p(J) is the polarization of compound nucleus after capture of the polarized neutron; B(Q, Λ, H) is a cumbersome factor
containing a bilinear combination of the amplitudes of fission to the exit states with two fragments.
Clearly, the term in (15) corresponding to Q = 1, Λ = H = 2 describes the required 5-fold T-odd
angular correlation
φ221 (nγ , nLF , n J ) ∼ (nγ [n J × nLF ])(nγ nLF ).
(16)
It is non-trivial that the factor B(Q, Λ, H) comprises the following product of three Klebsh-Gordan
Qh
JK 0
F0 K0
coefficients: C HhΛ0
C JKQh
C FKHh
. Here F and F 0 are total spins of two fragments, while K and K 0 are
projections of these spins onto the direction nLF of movement of the light fragment. At summation over
all exit channels it would be possible to expect a disappearance of the terms caused by an interference
Q0
= 0 at Q = 1, Λ = H = 2. Thus,
of the exit states with K , K 0 . But h = 0 when K = K 0 , and then C H0Λ0
the existence of the small (∼ 10−3 ) 5-fold T-odd angular correlation in binary fission is an evidence for
a regular mechanism of small mixing over K in all exit channels.
Our guess is that this mixing is connected with insignificant nonconservation of the quantum number K, caused by small (but nonzero) centrifugal barrier for the fission fragments. Therefore, the smallness of the R effect in binary fission points out the high accuracy of the matching condition K = K̃
at the scission point. Thus, the reliability of our treating of the A. Bohr channels in the frame of
Strutinsky’s approach is confirmed.
4 Conclusion
Processing the most comprehensive data on angular anisotropy of fragments from fission of spinaligned nuclei 235 U by slow (s-wave) neutrons shows that the A. Bohr’s channel K = 0 is not forbidden
for any J π but is relatively suppressed. It is due to specific symmetries of the internal wave function
related to nuclear forms on first and second fission barriers. The same data for the other target nuclei
are needed for consistent description of the K = 0 channel in connection with nuclear forms on fission
barriers.
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Recently discovered T- and R-effects (3- and 5-fold correlations) for the ”true” ternary fission
induced by slow polarized neutrons with α-particle’s emission can be caused by specific spin-orbit
interaction which mixes 3 particle’s exit states (final state interaction). On the other hand, the R-effect
for γ-quanta and neutrons, emitted by a fragment in binary fission induced by slow polarized neutrons,
provides a proof of correctness of the inclusion of A. Bohr’s channels into the helicity approach to the
description of angular correlations in the fission process.
This work was supported in part by Grants No. NS-7235.2010.2 and No. 2.1.1/4540 from MES of
Russia.
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